Definition df-finxp 32397

Description: Define Cartesian exponentiation on a class.

Note that this definition is limited to finite exponents, since it is defined using nested ordered pairs. If tuples of infinite length are needed, or if they might be needed in the future, use df-ixp 7795 or df-map 7746 instead. The main advantage of this definition is that it integrates better with functions and relations. For example if 𝑅 is a subset of (𝐴↑↑2_𝑜), then df-br 4584 can be used on it, and df-fv 5812 can also be used, and so on.

It's also worth keeping in mind that ((𝑈↑↑𝑀) × (𝑈↑↑𝑁)) is generally not equal to (𝑈↑↑(𝑀 +_𝑜 𝑁)).

This definition is technical. Use finxp1o 32405 and finxpsuc 32411 for a more standard recursive experience. (Contributed by ML, 16-Oct-2020.)

Assertion

Ref	Expression
df-finxp	⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}

Distinct variable groups:   𝑈,𝑛,𝑥,𝑦   𝑛,𝑁,𝑥,𝑦

Detailed syntax breakdown of Definition df-finxp

Step	Hyp	Ref	Expression
1		cU	. . 3 class 𝑈
2		cN	. . 3 class 𝑁
3	1, 2	cfinxp 32396	. 2 class (𝑈↑↑𝑁)
4		com 6957	. . . . 5 class ω
5	2, 4	wcel 1977	. . . 4 wff 𝑁 ∈ ω
6		c0 3874	. . . . 5 class ∅
7		vn	. . . . . . . 8 setvar 𝑛
8		vx	. . . . . . . 8 setvar 𝑥
9		cvv 3173	. . . . . . . 8 class V
10	7	cv 1474	. . . . . . . . . . 11 class 𝑛
11		c1o 7440	. . . . . . . . . . 11 class 1𝑜
12	10, 11	wceq 1475	. . . . . . . . . 10 wff 𝑛 = 1𝑜
13	8	cv 1474	. . . . . . . . . . 11 class 𝑥
14	13, 1	wcel 1977	. . . . . . . . . 10 wff 𝑥 ∈ 𝑈
15	12, 14	wa 383	. . . . . . . . 9 wff (𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈)
16	9, 1	cxp 5036	. . . . . . . . . . 11 class (V × 𝑈)
17	13, 16	wcel 1977	. . . . . . . . . 10 wff 𝑥 ∈ (V × 𝑈)
18	10	cuni 4372	. . . . . . . . . . 11 class ∪ 𝑛
19		c1st 7057	. . . . . . . . . . . 12 class 1st
20	13, 19	cfv 5804	. . . . . . . . . . 11 class (1st ‘𝑥)
21	18, 20	cop 4131	. . . . . . . . . 10 class ⟨∪ 𝑛, (1st ‘𝑥)⟩
22	10, 13	cop 4131	. . . . . . . . . 10 class ⟨𝑛, 𝑥⟩
23	17, 21, 22	cif 4036	. . . . . . . . 9 class if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)
24	15, 6, 23	cif 4036	. . . . . . . 8 class if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))
25	7, 8, 4, 9, 24	cmpt2 6551	. . . . . . 7 class (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
26		vy	. . . . . . . . 9 setvar 𝑦
27	26	cv 1474	. . . . . . . 8 class 𝑦
28	2, 27	cop 4131	. . . . . . 7 class ⟨𝑁, 𝑦⟩
29	25, 28	crdg 7392	. . . . . 6 class rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)
30	2, 29	cfv 5804	. . . . 5 class (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)
31	6, 30	wceq 1475	. . . 4 wff ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)
32	5, 31	wa 383	. . 3 wff (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))
33	32, 26	cab 2596	. 2 class {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
34	3, 33	wceq 1475	1 wff (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}

This definition is referenced by:  dffinxpf  32398  finxpeq1  32399  finxpeq2  32400  csbfinxpg  32401  finxp0  32404  finxp1o  32405  finxpnom  32414